Simplified Numerical Model for the Study of Wave Energy Convertes (WECs)

Similar documents
Modelling and Identification of Radiation-force Models for Ships and Marine Structures

Hydrodynamic Forces on Floating Bodies

SEAKEEPING AND MANEUVERING Prof. Dr. S. Beji 2

2 nd SDWED Symposium Advances in Modelling of Wave Energy Devices - WP4

( ) Chapter 3: Free Vibration of the Breakwater. 3.1 Introduction

CIS 4930/6930: Principles of Cyber-Physical Systems

Time- vs. Frequency-domain Identification of Parametric Radiation Force Models for Marine Structures at Zero Speed

A DERIVATION OF HIGH-FREQUENCY ASYMPTOTIC VALUES OF 3D ADDED MASS AND DAMPING BASED ON PROPERTIES OF THE CUMMINS EQUATION

1. Froude Krylov Excitation Force

Balanced Truncation 1

THE EFFECT OF MEMORY IN PASSIVE NONLINEAR OBSERVER DESIGN FOR A DP SYSTEM

Parametric Time-domain models based on frequency-domain data

Robot Control Basics CS 685

Identification Methods for Structural Systems. Prof. Dr. Eleni Chatzi Lecture March, 2016

A Discussion About Seakeeping and Manoeuvring Models For Surface Vessels

ROBUST PASSIVE OBSERVER-BASED CONTROL FOR A CLASS OF SINGULAR SYSTEMS

Hydrodynamic analysis and modelling of ships

ECEN 420 LINEAR CONTROL SYSTEMS. Lecture 6 Mathematical Representation of Physical Systems II 1/67

Identification of Dynamic Models of Marine Structures from Frequency-domain Data Enforcing Model Structure and Parameter Constraints

Coupled Heave-Pitch Motions and Froude Krylov Excitation Forces

Solutions to homework assignment #3 Math 119B UC Davis, Spring = 12x. The Euler-Lagrange equation is. 2q (x) = 12x. q(x) = x 3 + x.

Chapter 2: Complex numbers

Problem Set 5 Solutions 1

Physics 443, Solutions to PS 1 1

Compressible degree of freedom (CDOF): A potential strategy for improving wave energy capture

Lecture Notes Multibody Dynamics B, wb1413

WAMIT-MOSES Hydrodynamic Analysis Comparison Study. JRME, July 2000

DYNAMIC CHARACTERISTICS OF OFFSHORE TENSION LEG PLATFORMS UNDER HYDRODYNAMIC FORCES

1 POTENTIAL FLOW THEORY Formulation of the seakeeping problem

Proceedings of the ASME 27th International Conference on Offshore Mechanics and Arctic Engineering OMAE2008 June 15-20, 2008, Estoril, Portugal

Nonlinear Observer Design for Dynamic Positioning

Linear-quadratic control problem with a linear term on semiinfinite interval: theory and applications

Introduction to Modern Control MT 2016

AB-267 DYNAMICS & CONTROL OF FLEXIBLE AIRCRAFT

Characteristics of a pitching wave absorber with rotatable flap

Autonomous system = system without inputs

1 (30 pts) Dominant Pole


Theory of Ship Waves (Wave-Body Interaction Theory) Quiz No. 2, April 25, 2018

Lecture 7. Please note. Additional tutorial. Please note that there is no lecture on Tuesday, 15 November 2011.

Variational construction of periodic and connecting orbits in the planar Sitnikov problem. Mitsuru Shibayama(Kyoto University)

Discrete and continuous dynamic systems

Identification Methods for Structural Systems

= m(0) + 4e 2 ( 3e 2 ) 2e 2, 1 (2k + k 2 ) dt. m(0) = u + R 1 B T P x 2 R dt. u + R 1 B T P y 2 R dt +

Design of a Heading Autopilot for Mariner Class Ship with Wave Filtering Based on Passive Observer

Algebraic Properties of Solutions of Linear Systems

Output Stabilization of Time-Varying Input Delay System using Interval Observer Technique

Iterative methods to compute center and center-stable manifolds with application to the optimal output regulation problem

Part 8: Rigid Body Dynamics

The Calculus of Vec- tors

13.42 LECTURE 13: FLUID FORCES ON BODIES. Using a two dimensional cylinder within a two-dimensional flow we can demonstrate some of the principles

Control strategies in OWC plants

A sufficient condition for the existence of the Fourier transform of f : R C is. f(t) dt <. f(t) = 0 otherwise. dt =

Homogeneous and particular LTI solutions

Lecture 38: Equations of Rigid-Body Motion

A Review of Wave-to-Wire Models for Wave Energy Converters

Seakeeping Models in the Frequency Domain

INHOMOGENEOUS LINEAR SYSTEMS OF DIFFERENTIAL EQUATIONS

MEROS Project. Technical Advances in Modeling and Control. Dr. Lotfi CHIKH February (33)

Prof. Krstic Nonlinear Systems MAE281A Homework set 1 Linearization & phase portrait

. D CR Nomenclature D 1

DESIGN OF LINEAR STATE FEEDBACK CONTROL LAWS

Generalized coordinates and constraints

Dynamics of Structures

State space control for the Two degrees of freedom Helicopter

MATH 280 Multivariate Calculus Fall Integrating a vector field over a curve

Linear Systems Theory

EL 625 Lecture 10. Pole Placement and Observer Design. ẋ = Ax (1)

Optimal locomotion at low Reynolds number

L2 gains and system approximation quality 1

Lecture 38: Equations of Rigid-Body Motion

Control Systems. Frequency domain analysis. L. Lanari

Aeroelasticity. Lecture 9: Supersonic Aeroelasticity. G. Dimitriadis. AERO0032-1, Aeroelasticity and Experimental Aerodynamics, Lecture 9

Propagation of EM Waves in material media

Robot Dynamics Lecture Notes. Robotic Systems Lab, ETH Zurich

A MULTI-BODY ALGORITHM FOR WAVE ENERGY CONVERTERS EMPLOYING NONLINEAR JOINT REPRESENTATION

Obstacle problems and isotonicity

Final Exam TTK 4190 Guidance and Control

Oscillations and Waves

ME 234, Lyapunov and Riccati Problems. 1. This problem is to recall some facts and formulae you already know. e Aτ BB e A τ dτ

Model Reduction for Unstable Systems

Course 241: Advanced Mechanics. Scholarship Questions

Physical Dynamics (SPA5304) Lecture Plan 2018

PHY 5246: Theoretical Dynamics, Fall Assignment # 10, Solutions. (1.a) N = a. we see that a m ar a = 0 and so N = 0. ω 3 ω 2 = 0 ω 2 + I 1 I 3

Trajectory Tracking of a Near-Surface Torpedo using Numerical Methods

Carleson measures and elliptic boundary value problems Abstract. Mathematics Subject Classification (2010). Keywords. 1.

Classical Mechanics III (8.09) Fall 2014 Assignment 3

Linear dynamical systems with inputs & outputs

CDS 101: Lecture 5-1 Reachability and State Space Feedback

Implementation of latching control in a numerical wave tank with regular waves

Fourier Analysis Overview (0B)

Small oscillations and normal modes

Nonlinear Control. Nonlinear Control Lecture # 24 State Feedback Stabilization

Introduction to Path Integrals

Math Ordinary Differential Equations

Single-track models of an A-double heavy vehicle combination

Birth and Death Processes. Birth and Death Processes. Linear Growth with Immigration. Limiting Behaviour for Birth and Death Processes

Modeling and Analysis of Dynamic Systems

Robotics. Control Theory. Marc Toussaint U Stuttgart

Application of state observers in attitude estimation using low-cost sensors

Transcription:

Simplified Numerical Model for the Study of Wave Energy Convertes (WECs) J. A. Armesto, R. Guanche, A. Iturrioz and A. D. de Andres Ocean Energy and Engineering Group IH Cantabria, Universidad de Cantabria October 18 th, 213

1 Introduction 2 Aproximation 3 Identification 4 Applications

The movement of every body can be given by 6 Degrees of Freedom (DOF), three displacements: Surge, x Sway, y Heave, z and three rotations: Roll, φ Pitch, θ Yaw, ψ

Cummins equation in time domain: t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (1)

Cummins equation in time domain: t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (1) where: x = [x, y, z, φ, θ, ψ] T is the array of DOFs.

Cummins equation in time domain: t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (1) where: x = [x, y, z, φ, θ, ψ] T is the array of DOFs. M is the inertia matrix of the floating body and A the infinity added mass.

Cummins equation in time domain: t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (1) where: x = [x, y, z, φ, θ, ψ] T is the array of DOFs. M is the inertia matrix of the floating body and A the infinity added mass. K is the transfern function.

Cummins equation in time domain: t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (1) where: x = [x, y, z, φ, θ, ψ] T is the array of DOFs. M is the inertia matrix of the floating body and A the infinity added mass. K is the transfern function. C is the hydrostatic matrix.

Cummins equation in time domain: t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (1) where: x = [x, y, z, φ, θ, ψ] T is the array of DOFs. M is the inertia matrix of the floating body and A the infinity added mass. K is the transfern function. C is the hydrostatic matrix. f e are the external forces applied over the body,...

Ogilvie equation in frequency domain: ω 2 (M + A(ω)) X(iω) + iωb(ω)x(iω) + CX(iω) = F e (iω) (2)

Ogilvie equation in frequency domain: ω 2 (M + A(ω)) X(iω) + iωb(ω)x(iω) + CX(iω) = F e (iω) (2) Ogilvie Relations: A(ω) = A 1 ω B(ω) = K(t) sin(ωτ)dτ (3) K(t) cos(ωτ)dτ (4)

Ogilvie equation in frequency domain: ω 2 (M + A(ω)) X(iω) + iωb(ω)x(iω) + CX(iω) = F e (iω) (2) Ogilvie Relations: A(ω) = A 1 ω B(ω) = The transfern function can be written as: K(t) = 2 π K(iω) = 2 π K(t) sin(ωτ)dτ (3) K(t) cos(ωτ)dτ (4) B(ω) cos(ωt)dω & A = ĺım A(ω) (5) ω K(t)e iωt dt = B(ω) + iω[a(ω) A ]. (6)

Data A and B is obtained by a Boundary Element Method (BEM) software WADAM/WAMIT for a given set of frequencies.

Data A and B is obtained by a Boundary Element Method (BEM) software WADAM/WAMIT for a given set of frequencies. This program also provides M, C and wave forces for the given set of frequencies.

Data A and B is obtained by a Boundary Element Method (BEM) software WADAM/WAMIT for a given set of frequencies. This program also provides M, C and wave forces for the given set of frequencies. Then, we know: A(ω) and B(ω) for ω in {ω i } N n=1. M and C. F w (ω) for ω in {ω i } N n=1.

Convolution integrals in Cummins equation are computationally expensive, then we want to replace them for something more efficient. t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (7)

Convolution integrals in Cummins equation are computationally expensive, then we want to replace them for something more efficient. t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (7) We want to replace every convolution integral by a state-space: α(t) = A i,j α(t) + B i,j ẋ j (t) y i,j (t) = C i,j α(t) (8)

Convolution integrals in Cummins equation are computationally expensive, then we want to replace them for something more efficient. t (M + A ) ẍ(t) + K(t τ)ẋ(τ)dτ + Cx(t) = f e (t) (7) We want to replace every convolution integral by a state-space: α(t) = A i,j α(t) + B i,j ẋ j (t) y i,j (t) = C i,j α(t) (8) Such that: y i,j (t) t k i,j (t τ)ẋ j (τ)dτ, i, j {1, 2,..., 6} (9)

One Degree of Freedom Cummins equation results: t (M + A ) z(t) + K(t τ)ż(τ)dτ + Cz(t) = f e (t) (1)

One Degree of Freedom Cummins equation results: t (M + A ) z(t) + K(t τ)ż(τ)dτ + Cz(t) = f e (t) (1) Which can be written as: (M + A ) z(t) + y c (t) + Cz(t) = f e (t) (11)

One Degree of Freedom Cummins equation results: t (M + A ) z(t) + K(t τ)ż(τ)dτ + Cz(t) = f e (t) (1) Which can be written as: (M + A ) z(t) + y c (t) + Cz(t) = f e (t) (11) Lets define k 1 (t) = z(t) y k 2 (t) = ż(t) k 1 (t) = k 2 (t) k 2 (t) = z(t) (12)

One Degree of Freedom Cummins equation results: t (M + A ) z(t) + K(t τ)ż(τ)dτ + Cz(t) = f e (t) (1) Which can be written as: (M + A ) z(t) + y c (t) + Cz(t) = f e (t) (11) Lets define k 1 (t) = z(t) y k 2 (t) = ż(t) k 1 (t) = k 2 (t) k 2 (t) = z(t) (12) (M + A ) k 2 (t) + y c (t) + Cz(t) = f e (t) (13)

One Degree of Freedom Cummins equation results: t (M + A ) z(t) + K(t τ)ż(τ)dτ + Cz(t) = f e (t) (1) Which can be written as: (M + A ) z(t) + y c (t) + Cz(t) = f e (t) (11) Lets define k 1 (t) = z(t) y k 2 (t) = ż(t) k 1 (t) = k 2 (t) k 2 (t) = z(t) (12) (M + A ) k 2 (t) + y c (t) + Cz(t) = f e (t) (13) k 2 (t) = y c(t) M + A C M + A z(t) + f e(t) M + A (14)

One Degree of Freedom Then, we have: α(t) = A c α(t) + B c ẋ(t) y c (t) = C c α(t) t k(t τ)x(τ)dτ (15)

One Degree of Freedom Then, we have: α(t) = A c α(t) + B c ẋ(t) y c (t) = C c α(t) t k(t τ)x(τ)dτ (15) k 2 (t) = k 1 (t) (16)

One Degree of Freedom Then, we have: α(t) = A c α(t) + B c ẋ(t) y c (t) = C c α(t) t k(t τ)x(τ)dτ (15) k 2 (t) = k 1 (t) (16) C c k 2 (t) = α(t) M + A C M + A x(t) + f e(t) M + A (17)

One Degree of Freedom Definning χ = [α, k 1, k 2 ] = [α, x, ẋ] it is possible to rewrite previous equations into an unique state-space: where: χ(t) = Âχ(t) + ˆBf e (t) y(t) = Ĉχ(t) (18) Â = A c. B c... 1 C c /a C/a ; ˆB = 1/a ; Ĉ = [... 1 ] Being a = M + A. We have written Cummins equation as an unique state-space! (19)

More than one Degree of Freedom The same procedure can be used for more than one degree of freedom. [ ξ]] T χ(t) = [α c1 ], [α c2 ],... [α cn ], [ξ], [ (2) Â = [A c1 ] [B c1 ]....... [A cn ] [B cn ]. [Id] [C c1 /M]... [C cn /M] [G/M] ; ˆB =. [ M 1] (21) Ĉ = [... 1 ] (22)

With everything we known, how do we compute the matrix and vectors that approximate the convolution integrals?

With everything we known, how do we compute the matrix and vectors that approximate the convolution integrals? Data: A(ω) and B(ω) for ω {ω i } N n=1.

With everything we known, how do we compute the matrix and vectors that approximate the convolution integrals? Data: A(ω) and B(ω) for ω {ω i } N n=1. There are two options: Identification in time domain: K(t) = 2 π B(ω) cos(ωt)dω (23)

With everything we known, how do we compute the matrix and vectors that approximate the convolution integrals? Data: A(ω) and B(ω) for ω {ω i } N n=1. There are two options: Identification in time domain: K(t) = 2 π or identificatio in frecuency domain: B(ω) cos(ωt)dω (23) K(iω) = 2 π K(t)e iωt dt = B(ω) + iω[a(ω) A ]. (24)

Using A(ω) and B(ω) it is possible to built the transfern function for every frequency ω {ω i } N n=1 : K(iω j ) = B(ω j ) + iω j (A(ω j ) A ) (25)

Using A(ω) and B(ω) it is possible to built the transfern function for every frequency ω {ω i } N n=1 : K(iω j ) = B(ω j ) + iω j (A(ω j ) A ) (25) A Matlab Toolbox done by Pérez and Fossen, compute values θ = [p, p 1,..., p n 1, q, q 1,..., q n 1 ] such that: ˆK(s, θ) = p n 1s n 1 + p n 2 s n 2 +... + p 1 s 1 + p s n + q n 1 s n 1 + q n 2 s n 2 +... + q 1 s 1 + q (26) is the best possile approximation (least square sense) for K(iω), ω {ω i } N n=1

Using A(ω) and B(ω) it is possible to built the transfern function for every frequency ω {ω i } N n=1 : K(iω j ) = B(ω j ) + iω j (A(ω j ) A ) (25) A Matlab Toolbox done by Pérez and Fossen, compute values θ = [p, p 1,..., p n 1, q, q 1,..., q n 1 ] such that: ˆK(s, θ) = p n 1s n 1 + p n 2 s n 2 +... + p 1 s 1 + p s n + q n 1 s n 1 + q n 2 s n 2 +... + q 1 s 1 + q (26) is the best possile approximation (least square sense) for K(iω), ω {ω i } N n=1 This means that this toolbox provides element θ that minimizes the following operatos: θ = mín θ ( n ) ( K(iω n ) ˆK(iω ) 2 n, θ) (27)

Properties the identification must : ĺım ˆK(iω) = It has ceros for s = (28) ω

Properties the identification must : ĺım ˆK(iω) = It has ceros for s = (28) ω ĺım ˆK(iω) = It is strictly proper (29) ω

Properties the identification must : ĺım ˆK(iω) = It has ceros for s = (28) ω ĺım ˆK(iω) = It is strictly proper (29) ω ĺım ˆK(t) Relative order 1 (gr(q) = gr(p)+1) (3) t +

Properties the identification must : ĺım ˆK(iω) = It has ceros for s = (28) ω ĺım ˆK(iω) = It is strictly proper (29) ω ĺım ˆK(t) Relative order 1 (gr(q) = gr(p)+1) (3) t + ĺım ˆK(t) = BIBO Stable (31) t

Properties the identification must : ĺım ˆK(iω) = It has ceros for s = (28) ω ĺım ˆK(iω) = It is strictly proper (29) ω ĺım ˆK(t) Relative order 1 (gr(q) = gr(p)+1) (3) t + ĺım ˆK(t) = BIBO Stable (31) t Ẋ y is passive Re( ˆK(iω)) > (32)

Once we know θ = [p, p 1,..., p n 1, q, q 1,..., q n 1 ]:... q 1... q 1 A c = 1... q 2...... ; B c = p p 1 p 2. ; (33)... 1 q n 1 p n 1 C c = [,,...,, 1] y c (t) = CX c (t) = x n (t) (34).

CatAir

CatAir

CatAir t (M + A ) z(t)+ K(t τ)z(τ)dτ +Cz(t)+k l ż+k nl ż ż = f e (t) (35)

CatAir t (M + A ) z(t)+ K(t τ)z(τ)dτ +Cz(t)+k l ż+k nl ż ż = f e (t) (35) t (M + A ) z(t)+ K(t τ)z(τ)dτ+cz(t)+(k l + k nl ż ) ż = f e (t) (36)

CatAir t (M + A ) z(t)+ K(t τ)z(τ)dτ +Cz(t)+k l ż+k nl ż ż = f e (t) (35) t (M + A ) z(t)+ K(t τ)z(τ)dτ+cz(t)+(k l + k nl ż ) ż = f e (t) (36) (M + A ) k 2 (t) + y c (t) + Cz(t) + (k l + k nl ż) ż = f e (t) (37) k 2 (t) = C c C X c (t) z(t) k l + k nl ż ż + f e(t) (38) M + A M + A M + A M + A

CatAir We have defined X (t) such that X n (t) = ż(t), then: A = A c. B c... 1 C c /a b/a (k l + k nl X n )/a ; B = 1/a (39) C = [... 1 ] (4)

CatAir

CatAir ṁ air = d dt (ρ air (t)v air (t)); V air (t) = V Sz(t) = S(L z(t)). (41) Assuming the proccess is isentropic, ρ air (t)(p + P(t)) 1/γ = K The air mass flux through the openning can be expressed as follows: [ ] 2mair (t) P(t) 1/2 ṁ air (t) = c d A v sgn( P(t)) (42) V air (t)

CatAir Lets define the relative pressure P (t) = P(t)/P = ( P(t) + P )/P. The air pressure variation inside the chamber, Ṗ (t) = [ A v c d γk ( ) v 1/2 sgn(p (t) 1) 2P ρ P (t) 1/γ P (t) 1 ρ S Cummins equation as solved using a state-space, +γż(t)] P (t) = ΓP (t) (43) Ẋ (t) = AX (t) + Bu(t) y(t) = CX (t) Cummins equation is modified to include pressure effects, (44) (m + A ) z(t) + y c (t) + ρgsz(t) + kż(t) P(t)S = f e (t) (45)

CatAir Writting together equations (45) and (43), we can write a new state-space. The first step is to redefine the space s vector, X (t) = [ X (t) P (t) ]. (46) Then matrix A and arrays B and C are redefined as: A = A S... Γ ; B = B ; C = [ C ]. (47)

CatAir Figura: Laboratory and time-domain model free surface and chamber dynamic pressure time series. Regular waves (H =,8 m, T = 1,1 s), top opening=4.5 mm, c d =,8643

WEFIES P = C PTO (ż 1 ż 2 ) 2.

WEFIES Laboratory tests coming (hopefully) soon...

WEFIES Thanks for your attention!

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback